{
"cells": [
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "slide"
}
},
"source": [
"# 1.2 Quelques représentations de fonctions $f : \\mathcal{U} \\subseteq \\mathbb{R}^m \\to \\mathbb{R}^n$"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "fragment"
}
},
"source": [
"## Cas 1 : courbes et chemins\n",
"\n",
"Un chemin est une fonction $\\mathbf{\\gamma} : I \\subseteq \\mathbb{R} \\to \\mathbb{R}^n$. Voici plusieurs exemples faits avec SageMath.\n",
"\n",
"**Exemple b)** des notes : $\\gamma(t) = (t\\cos{t}, t\\sin{t}, t), t\\in \\mathbb{R}$."
]
},
{
"cell_type": "code",
"execution_count": 1,
"metadata": {
"slideshow": {
"slide_type": "slide"
}
},
"outputs": [
{
"data": {
"text/html": [
"\n",
"\n"
],
"text/plain": [
"Graphics3d Object"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"var('t')\n",
"Courbe = parametric_plot3d((t*cos(t), t*sin(t),t), # le chemin\n",
" (t,-pi, 4*pi), # le domaine du parametre\n",
" frame = True, # la boite\n",
" color= \"red\")\n",
"show(Courbe)"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "subslide"
}
},
"source": [
"On l'a vu, cette courbe se trouve sur le cône $x^2+y^2 = z^2$. On va créer le nouvel élément de la figure totale, puis les additionner."
]
},
{
"cell_type": "code",
"execution_count": 2,
"metadata": {
"slideshow": {
"slide_type": "fragment"
}
},
"outputs": [
{
"data": {
"text/html": [
"\n",
"\n"
],
"text/plain": [
"Graphics3d Object"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"var('x,y,z')\n",
"Cone = implicit_plot3d(x^2 +y^2 - z^2 ==0, (x,-4*pi,4*pi), (y,-4*pi,4*pi), (z,-pi,4*pi),\n",
" color = \"orange\",\n",
" opacity = 0.65,\n",
" frame = False,\n",
" viewer = \"threejs\")\n",
"show(Cone +Courbe)"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "slide"
}
},
"source": [
"**Exemple c)** des notes.\n",
"On s'intéresse à la surface $\\{ z^2 = x^2 + y^2\\}\\cap \\{z = 1+y\\}$"
]
},
{
"cell_type": "code",
"execution_count": 3,
"metadata": {
"slideshow": {
"slide_type": "subslide"
}
},
"outputs": [
{
"data": {
"text/html": [
"\n",
"\n"
],
"text/plain": [
"Graphics3d Object"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"var('t')\n",
"C1 = parametric_plot3d([sqrt(1+2*t), t, 1+t], (t,-1/2,4*pi), \n",
" color= \"blue\", thickness= 2)\n",
"C2 = parametric_plot3d([-sqrt(1+2*t), t, 1+t], (t,-1/2,4*pi), \n",
" color= \"green\", thickness = 2)\n",
"show(C1+C2+Cone)"
]
},
{
"cell_type": "code",
"execution_count": 4,
"metadata": {
"slideshow": {
"slide_type": "slide"
}
},
"outputs": [
{
"data": {
"text/html": [
"\n",
"\n"
],
"text/plain": [
"Graphics3d Object"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"var('s')\n",
"C3 = parametric_plot3d([s, (s^2 - 1)/2, (s^2 +1)/2], (s,-2*pi, 2*pi), \n",
" color = \"red\", thickness = 2)\n",
"show(Cone +C3)"
]
},
{
"cell_type": "code",
"execution_count": 5,
"metadata": {
"slideshow": {
"slide_type": "subslide"
}
},
"outputs": [
{
"data": {
"text/html": [
"\n",
"